If 1 woman is randomly selected, the probability that her height is less than 63 in will be:
The z-score is given by:
z=(x-mu)/sig
z=(63-62.5)/2.6
z=0.1923
Thus:
P(x<63)=P(z<0.1923)=0.5753
(b) if 44 women are randomly selected, find the probability that they have a mean height less than 63 in.
In this scenario the desired probability is for the mean of a sample of 44 women, therefore we use the central limit theorem. The standard deviation of the sample means is:
2.6/sqrt44=0.392
since n>30, use the z-distribution
z=(63-62.5)/0.392
z=1.28
Then
P(x-bar <63)=P(z<1.28)=0.8997
